Our main algorithm replaces the sequential procedure from [1]} to decide
if a bounded clique-width graph has a Hamiltonian path. Instead, it computes
the number of Hamiltonian cycles. The procedure uses elementary counting coupled
with polynomial interpolation to compute some matrices which are independent
of the input graph depending only on its size.
The matrices are then combined with vectors maintaining counts, along the
structure tree of the clique-decomposition. A degree bound for the monotone
arithmetic circuit then suffices to prove the #SAC\(^1\) bound.
